Math problem complex numbers and trigonometric function. Please help thank you.?

Question

Find z1z2 and z1/z2 as complex numbers in trigonometric form. 

z1=2+10i 
z2=-4-5i

Answer 1

1).

The polar form(trigonometric form) of a complex number z = a + bi is z = r (cos θ + i sin θ), where r = | z | = √(a² + b²), a = r cos θ, and b = r sin θ, and θ = tan^{- 1}(b / a) for a > 0 or θ = tan^{- 1}(b / a) + π or θ = tan^{- 1}(b / a) + 180^o for a < 0.

The complex numbers are z1 = 2 + 10i and z2 = - 4 - 5i.

z1*z2 = (2 + 10i)(- 4 - 5i)

= - 8 - 40i - 10i - 50i²

Substitute the value - 1 for i².

= - 8 - 50i - 50(- 1)

= - 8 - 50i + 50

= 42 - 50i.

z1z2 = 42 - 50i.

The polar form of a complex number z = a + bi is z = r (cos θ + i sin θ).

Here a = 42 > 0 and b = - 50.

So, first find the absolute value of r .

r = | z | = √(a² + b²)

            = √[ (42)² + (- 50)² ]

            = √[ 1764 + 2500 ]

              = √4264

            = 65.23.

Now find the argument θ.

Since a = 42 > 0, use the formula θ = tan^{- 1}(b / a).

θ = tan^{- 1}[ - 50/42 ]

θ = tan^{- 1}(- 1.19)

θ = - 49.96.

Note that here θ is measured in degrees.

Therefore, the trigonometric form of z1z2 is 65.23[ cos (- 49.96^o) + i sin (- 49.96^o) ].

Answer 2

2).

z1 / z2 = (2 + 10i) / (- 4 - 5i).

Multiply the numerator and denominator by conjugate of - 4 - 5i.

z1 / z2 = (2 + 10i)(- 4 + 5i) / (- 4 + 5i)(- 4 - 5i)

= [- 8 - 40i + 10i + 50i²] / [(- 4)² - (5i)²)

= [- 8 - 30i - 50] / [16 - (25i²)]

= [- 8 - 30i - 50] / [16 + 25]

= [- 58 - 30i] / 41

z1 / z2 = (- 58/41) - (30/41)i.

Here a = - 58/41 < 0 and b = - 30/41.

So, first find the absolute value of r .

r = | z | = √(a² + b²)

            = √[ (- 58/41)² + (- 30/41)² ]

            = √[ (3364/1681) + (900/1681) ]

            = √4264/41

            = 65.23/41

            = 1.59.

Now find the argument θ.

Since a = - 58/41 < 0, use the formula θ = tan^{- 1}(b / a) + π.

θ = tan^{- 1}[ 30/58 ] + 180

θ = tan^{- 1}(0.517) + 180

θ = 27.34 + 180

θ = 207.34.

Note that here θ is measured in degrees.

Therefore, the trigonometric form of z1/z2 is 1.59[ cos (207.34^o) + i sin (207.34^o) ].